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The distribution of random parameters in, and the input signal to, adistributed parameter model with unbounded input and output operators for thetransdermal transport of ethanol are estimated. The model takes the form of adiffusion equation with the input, which is on the boundary of the domain,being the blood or breath alcohol concentration (BAC/BrAC), and the output,also on the boundary, being the transdermal alcohol concentration (TAC). Ourapproach is based on the reformulation of the underlying dynamical system insuch a way that the random parameters are treated as additional spatialvariables. When the distribution to be estimated is assumed to be defined interms of a joint density, estimating the distribution is equivalent toestimating a functional diffusivity in a multi-dimensional diffusion equation.The resulting system is referred to as a population model, and well-establishedfinite dimensional approximation schemes, functional analytic based convergencearguments, optimization techniques, and computational methods can be used tofit it to population data and to analyze the resulting fit. Once the forwardpopulation model has been identified or trained based on a sample from thepopulation, the resulting distribution can then be used to deconvolve theBAC/BrAC input signal from the biosensor observed TAC output signal formulatedas either a quadratic programming or linear quadratic tracking problem. Inaddition, our approach allows for the direct computation of correspondingcredible bands without simulation. We use our technique to estimate bivariatenormal distributions and deconvolve BAC/BrAC from TAC based on data from apopulation that consists of multiple drinking episodes from a single subjectand a population consisting of single drinking episodes from multiple subjects.

inverse problems

Deconvolving the Input to Random Abstract Parabolic Systems; A  Population Model-Based Approach to Estimating Blood/Breath Alcohol  Concentration from Transdermal Alcohol Biosensor Data